Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 2x - 13$, and $ m \angle LOM = 5x - 37$, find $m\angle MON$. $O$ $L$ $N$ $M$
From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {5x - 37} + {2x - 13} = {90}$ Combine like terms: $ 7x - 50 = 90$ Add $50$ to both sides: $ 7x = 140$ Divide both sides by $7$ to find $x$ $ x = 20$ Substitute $20$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 2({20}) - 13$ Simplify: $ {m\angle MON = 40 - 13}$ So ${m\angle MON = 27}$.